Aircraft: B737
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Critical; Inflight Event / Encounter Unstabilized Approach

After takeoff; the FO who was flying; seemed to be having trouble stabilizing the aircraft in roll. He kept working with the aileron and rudder trim. We elected to stop climbing upon reaching 15000 feet in order to analyze the problem. If the rudder and aileron trim were set back to '0' you had to hold the control wheel right wing down at 5 units according to the control wheel scale. If you let go; the aircraft rolled left at about 10 degrees per second. I elected to go into the cabin and see what I could see and the only thing I saw was the right spoiler slightly lifted as I expected. I did not look to see if all the trailing edges were even as the flap indicator showed full up and even pointers. After considerable difficulty of getting and maintaining communication with commercial radio were established. After discussing the problem with maintenance control; we elected to [advise ATC] and return to ZZZ as the best course of action. I told the flight attendants I had expected a normal landing and no evacuation. I informed the passengers we were returning to ZZZ and that they would see fire trucks and that was normal in the interest of extra precaution. After we began extending flaps for landing; the roll tendency diminished until reaching flaps 30. At which the FO said it seemed as though the aircraft wanted to roll to the right a little. We landed overweight with a smooth touchdown. After we got to the gate; a mechanic came to the flight deck and said that they had found the problem; the right inboard trailing edge flap had a broken actuator and was extended. All flight deck indicators indicated normal and proper indications at all times.Since the flap pointers were perfectly even; displaying full up; flap asymmetry just did not occur to us. There were no indications of any problem except the pronounced uncommanded roll. I told the FO that I wanted to go back and look at the wing. He opposed going into the cabin citing security concerns. He suggested having a flight attendant look. Personally; I felt like I would not get good information from a flight attendant in this situation. I called the FAs and they reported everything seemed normal to them. When I asked for someone to go look at the wing; they only handed the phone from one to another. So I told the FO I was going to go back and look and he concurred. Since the flap indicator was showing even needles and full up; I believe I had a preconceived expectation that the flaps were not the problem so I did not check them. I only looked at the spoilers mainly and they looked as expected. All it did was to further place in my mind that it was not a flap; slat or spoiler problem. I was hoping maintenance control would have another idea; however it seemed maintenance control was more interested in trying to get us to just use a combination of aileron and rudder trim to get the aircraft on down the road. After listening to him for a while and complying with his request; it became clear to me that his viewpoint was that this was a minor issue. I felt like the dispatcher did grasp the situation. We [advised ATC] and returned to ZZZ which were not far from. The FO was the PF and he did an excellent job throughout the flight and in all aspects of CRM. B737 Captain reported being unable to stabilize the aircraft on its roll axis. Setting the aileron and rudder trim to zero; the pilot flying had to hold the control wheel right wing down five units; according to the control wheel scale; to stabilize the aircraft. After landing Maintenance informed the crew of the right inboard flap actuator being broken.